Numerical Analysis/Matrix norm exercises

Consider the general $$3\times 3$$ matrix $$ A=     \begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \\ \end{bmatrix} $$ and the specific example $$ B=     \begin{bmatrix} 1 & 2 & 3 \\          4 & 5 & 6 \\           7 & 8 & 9 \\      \end{bmatrix} $$.

Find $$\left \| A \right \| _1 $$ Solution: given: $$ \left \| A \right \| _1 = \max \limits _{1 \leq j \leq n} \sum _{i=1} ^n | a_{ij} | $$

Solve for column 1: $$\sum _{i=1}^3|a_{i1}| = |a_{11}|+|a_{21}|+|a_{31}|$$

Solve for column 2: $$\sum _{i=1}^3|a_{i2}| = |a_{12}|+|a_{22}|+|a_{32}|$$

Solve for column 3: $$\sum _{i=1}^3|a_{i3}| = |a_{13}|+|a_{23}|+|a_{33}|$$

$$ \left \| A \right \| _1$$ = the maximum absolute column sum of the matrix

Find $$\left \| B \right \| _1$$ Solution: Solve for column 1: $$\sum _{i=1}^3|b_{i1}| = |1|+|4|+|7| = 12 $$

Solve for column 2: $$\sum _{i=1}^3|b_{i2}| = |2|+|5|+|8| = 15 $$

Solve for column 3: $$\sum _{i=1}^3|b_{i3}| = |3|+|6|+|9| = 18 $$

$$ \left \| B \right \| _1 = \max[12,15,18] = 18 $$

Find the infinite norm, $$\left \| A \right \| _\infty$$ Solution: given: $$ \left \| A \right \| _\infty = \max \limits _{1 \leq i \leq n} \sum _{j=1} ^n | a_{ij} | $$

Solve for row 1: $$\sum _{j=1}^3|a_{1j}| = |a_{11}|+|a_{12}|+|a_{13}|$$

Solve for row 2: $$\sum _{j=1}^3|a_{2j}| = |a_{21}|+|a_{22}|+|a_{23}|$$

Solve for row 3: $$\sum _{j=1}^3|a_{3j}| = |a_{31}|+|a_{32}|+|a_{33}|$$

$$ \left \| A \right \| _\infty$$ = the maximum absolute row sum of the matrix

Find the infinite norm, $$\left \| B \right \| _\infty$$ Solution: Solve for row 1: $$\sum _{j=1}^3|b_{1j}| = |1|+|2|+|3| = 6 $$

Solve for row 2: $$\sum _{j=1}^3|b_{2j}| = |4|+|5|+|6| = 15 $$

Solve for row 3: $$\sum _{j=1}^3|b_{3j}| = |7|+|8|+|9| = 24 $$

$$ \left \| B \right \| _\infty = \max[6,15,24] = 24 $$

Find the Frobenius norm, $$\left \| A \right \| _F$$ Solution: given: $$\|A\|_F=\sqrt{\sum_{i=1}^3\sum_{j=1}^3 |a_{ij}|^2}$$

$$\|A\|_F=\sqrt{|a_{11}|^2+|a_{12}|^2+|a_{13}|^2+|a_{21}|^2+|a_{22}|^2+|a_{23}|^2+|a_{31}|^2+|a_{32}|^2+|a_{33}|^2}$$

Find the Frobenius norm, $$\left \| B \right \| _F$$ Solution: $$\|B\|_F=\sqrt{|1|^2+|2|^2+|3|^2+|4|^2+|5|^2+|6|^2+|7|^2+|8|^2+|9|^2} = \sqrt{285} \approx 16.9 $$